| Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
| Name: | $C_3$ |
| Order: | $3$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$62$ |
$0$ |
$1105$ |
$0$ |
$23954$ |
$0$ |
$582246$ |
$0$ |
$15203628$ |
| $a_2$ |
$1$ |
$3$ |
$17$ |
$125$ |
$1101$ |
$10933$ |
$117631$ |
$1336779$ |
$15789869$ |
$191887373$ |
$2383264131$ |
$30116329675$ |
$385978635295$ |
| $a_3$ |
$1$ |
$0$ |
$24$ |
$0$ |
$2728$ |
$0$ |
$527330$ |
$0$ |
$126636216$ |
$0$ |
$34000066212$ |
$0$ |
$9777495522696$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$17$ |
$10$ |
$31$ |
$62$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$24$ |
$125$ |
$74$ |
$249$ |
$148$ |
$518$ |
$1105$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$186$ |
$1101$ |
$644$ |
$380$ |
$2317$ |
$1356$ |
$4986$ |
$2910$ |
$10875$ |
$23954$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1720$ |
$10933$ |
$1004$ |
$6320$ |
$3668$ |
$23793$ |
$13732$ |
$7952$ |
$52394$ |
$30170$ |
$116285$ |
$66808$ |
$259602$ |
$582246$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2728$ |
$17604$ |
$117631$ |
$10148$ |
$67292$ |
$38596$ |
$261349$ |
$22194$ |
$149224$ |
$85402$ |
$584498$ |
$333080$ |
$190236$ |
$1313469$ |
| $$ |
$747124$ |
$2962638$ |
$1682352$ |
$6702822$ |
$15203628$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&2&0&5&0&0&8\\0&5&0&5&0&16&0&0&17&0&10&0&22&37&0\\2&0&12&0&12&0&26&30&0&16&0&48&0&0&88\\0&5&0&9&0&24&0&0&31&0&14&0&44&65&0\\2&0&12&0&17&0&31&36&0&22&0&68&0&0&116\\0&16&0&24&0&88&0&0&112&0&60&0&168&252&0\\3&0&26&0&31&0&82&90&0&62&0&163&0&0&328\\4&0&30&0&36&0&90&108&0&72&0&188&0&0&384\\0&17&0&31&0&112&0&0&169&0&76&0&250&373&0\\2&0&16&0&22&0&62&72&0&60&0&132&0&0&296\\0&10&0&14&0&60&0&0&76&0&53&0&127&188&0\\5&0&48&0&68&0&163&188&0&132&0&373&0&0&728\\0&22&0&44&0&168&0&0&250&0&127&0&401&580&0\\0&37&0&65&0&252&0&0&373&0&188&0&580&877&0\\8&0&88&0&116&0&328&384&0&296&0&728&0&0&1600\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&9&17&88&82&108&169&60&53&373&401&877&1600&713&747&1647&1280&327\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
